My original intent was to enable push support for my mail server and the following recommendation works pretty well for dozens of mails per hour with usually brilliant, rarely not-so-jolly response times.
There are a couple of choices in the App Store. The first I found bearable is Boxcar. It is an app coupled with a service that provides you access to Apple Push Notification Service in a form of a unique e-mail address which you can forward mail to.
It has a variety of notifications, but the best idea is to set it to merely display badges with mail information and play a sound.
The downside is that you have to pipe your mail through a third party but you have a choice to filter forwarded mail on your own mail server, which you already do.
I find it blazing fast most of the time (95%) with response times measured in sub-4 seconds, but there are cases (5%) where the delivery lag jitters highly. I have seen messages appear on the desktop computer 15 minutes before the push arrived.
Still, it's easy to configure and it is actually convenient.